Differential Sampling Applied to Mammography Image Simulationpeplowde/p013.pdf · Mammography is...

Differential Sampling Applied to Mammography Image Simulation

Douglas E. Peplow*

Oak Ridge National LaboratoryBuilding 6025 Mailstop 6363, P.O. Box 2008

Oak Ridge, Tennessee 37831-6363

and

Kuruvilla Verghese

North Carolina State UniversityDepartment of Nuclear Engineering

Raleigh, North Carolina 27695

Received June 1, 1999Accepted November 23, 1999

Abstract –Differential sampling is a powerful tool that allows Monte Carlo to compute derivatives ofresponses with respect to certain problem parameters. This capability has been implemented within anin-house Monte Carlo code that simulates detailed mammographic images from two new digital systems.Differential sampling allows for the calculation of the first and all second derivatives of all of the differenttallies computed by the code as well as the first and second derivatives of the mammographic image itselfwith respect to material parameters, such as density and cross sections. The theory behind differentialsampling is explained, the methodology for implementation into the imaging code is discussed, and twoproblems are used to demonstrate the power of differential sampling.

I. INTRODUCTION

Mammography is the X-ray radiographic techniquethat is designed for breast imaging. The lifetime risk ofdeveloping breast cancer is one out of nine for women inthe United States, and the earlier the tumor is detected,the better is the prognosis for survival. X-ray mammog-raphy has enabled early detection of tumors and has ledto a large increase in survivability. At present, annualmammograms are indicated for all women more than 50 yr~postmenopausal! and for high-risk women in youngerage groups. Contrast in the images is greater in the higherage groups because of the higher adipose tissue contentof their breasts, as compared to premenopausal womenwho have more glandular tissue than adipose. Although

advances in mammography have resulted in image con-trast enhancement, the goal of mammography is to im-prove the detection of smaller tumors in premenopausalwomen without increasing the radiation dose. Methodsthat do not use radiation, such as ultrasound and mag-netic resonance imaging, are at present not competitivewith X-ray mammography for routine cancer screening.

Mammogram images are formed just like standardX-ray images. A beam of X rays is passed through thetissue and then strikes a film. Areas of higher densitywill absorb more X rays~transmitting less!, resulting ina lower exposure to the film. If X rays were only eithertransmitted or absorbed, the image captured by the filmwould perfectly show which areas of the tissue ab-sorbed more photons. Unfortunately, X rays can alsoscatter in the tissue. Scattered X rays can strike the filmin any location, adding a constant level of exposure tothe entire image.*E-mail: [email protected]

NUCLEAR SCIENCE AND ENGINEERING:135, 103–122~2000!

103

Standard clinical mammography systems use anX-ray tube bremsstrahlung spectrum that is filtered to re-move very low energy photons. A conical X-ray beam istransmitted through the breast that is compressed be-tween two flat plastic paddles. The transmitted beampasses through an antiscatter grid to a phosphor screen,and the image is recorded on photographic film. The gridremoves some of the scattered X rays but increases theamount of source photons needed and increases the ab-sorbed dose in the breast. The tube anode potential is ad-justed to suit the breast thickness.

Image contrast is degraded by scattered radiationstriking the detector and geometric blurring by nonpointsources and diverging beams, and by detection systemsthemselves. The high-energy portion of a tube anode spec-trum does not add much information to the image be-cause of low-interaction cross sections. The low-energyportion of the spectrum only increases dose to the breastsince low-energy photons have a very high interactioncross section and are rarely transmitted. Only the mid-range of photon energies contributes to the contrast seenin the images. The optimum photon energy range for com-pressed breast thicknesses of 3 to 6 cm is;17 to 25 keV.

There have been both evolutionary and revolution-ary responses to the need to enhance mammography tothe point of detecting smaller tumor masses earlier. Im-provements in film quality, scatter rejection grids, bet-ter digital detectors, and better computer imageenhancements are all leading to clearer mammogramsin this country. Many manufacturers of current mam-mography machines are developing new systems thathave digital data acquisition and scanning slots to fur-ther decrease the amount of scattered radiation—all withthe goal of an improved image. These systems still usethe tube anode and a bremsstrahlung spectrum of X raysbut are improvements over the conventional film0screen systems.

Research groups in this country and abroad are work-ing on completely new mammography technologies usingintense monoenergetic beams of X-ray radiation from syn-chrotron sources.1–3 Images produced from the monoen-ergetic, parallel, polarized synchrotron radiation showbetter contrast than conventional images, approaching thetheoretical limit for an X-ray attenuation image. With thetunable monoenergetic beam, a clear optimization can bemade for contrast and dose. The parallel aspect of syn-chrotron radiation allows for large air gaps between thepatient and detector, greatly reducing the effect of scatterwithout any magnification or geometric blurring.

This work stems from a collaboration with the groupthat started research in synchrotron mammography.4 Thegoal of the project is to develop a set of modeling toolsthat could be used to compare the performance ofthe new synchrotron system to the next generation ofscanning-slot mammography systems that are being testedin clinics and hospitals. Sensitivities to various materialparameters and perturbation calculations are of interest

to this group and others in medical physics. The longerrange goal is to use these modeling tools to optimize newmammography systems.

The best way to quantify the effects of scattered ra-diation in the images from different systems and also com-pare other system effects is to create a Monte Carlo toolcapable of simulating the complete image. Digital mam-mograms are typically 20483 2048 pixels in size as aminimum. Tallying photons into more than four millionpixels poses some challenges to any Monte Carlo codebecause of the potentially high variance in each of thepixels. A combination of variance-reduction methodsmade it possible to simulate accurate images using realpixel dimensions within reasonable computation times.The complete method of image simulation, including theapplication of noise and the modulation transfer function~MTF!, both of which tend to smear images, has beenpresented by the authors elsewhere.5 Simulated imagesof two common mammography phantoms on both thedigital Fischer Senoscan scanning-slot system and thesynchrotron-based mammography system showed excel-lent agreement with actual digital images taken on thetwo systems.

This paper will focus on differential sampling andhow it was applied to the image simulation problem. Usingdifferential sampling, Monte Carlo can calculate not onlythe desired response but also the derivative of that re-sponse to some parameter. These derivatives can be usedto determine sensitivities of dose-to-material param-eters or sensitivity of image contrast to tumor density.They can also be used to propagate uncertainties in thematerial data to find the final dose uncertainty. With atruncated Taylor series, the derivatives can be used in per-turbation studies. A demonstration of how differentialsampling can be used to provide more~and useful! in-formation for imaging problems appears in Sec. V.

II. THE MCMIS MONTE CARLO IMAGING CODE

The Monte Carlo Mammography Image Simula-tion5 ~MCMIS! is a detailed code specifically for the sim-ulation of digital mammography systems, includingscanning-slot systems. This code was written so that threevariance-reduction techniques can be used together to cre-ate images with realistic pixel sizes. These are source ras-tering, the separation of the scattered and unscatteredimage, and the point-detector scheme.6 The source ras-tering is just a form of stratified sampling and ensuresthat the images do not suffer from mottle caused by ran-dom source sampling. The point-detector method forcesa score to every pixel in the image at each photon inter-action, thus decreasing the variance of the final scatteredimage. Since the scattered image does not show the de-tail that the unscattered images show, it can be calcu-lated on a coarse mesh, saving computation time.

104 PEPLOW and VERGHESE

NUCLEAR SCIENCE AND ENGINEERING VOL. 135 JUNE 2000

In the energy range of medical imaging, the coher-ent scattering distribution from amorphous materials, suchas tissues and plastics, differs a great deal from thescattering distributions predicted by the free-gas model.7

One example is shown in Fig. 1. Both of the mammog-raphy systems studied here are well collimated, whicheliminates photon scatter through more than a few de-grees from the image. In the small-angle range, the dif-ference between the measured and predicted coherentscatter distributions is very large, and the amount of in-coherent scatter is small. To make the simulation of mam-mography systems as complete as possible, we measuredthe molecular coherent scattering form factors of mam-mography related materials.8 These measurements useda unique two-step approach that accurately accounted forincoherent scatter and multiple scatter. The measure-ments were performed using monoenergetic synchrotronradiation, which makes the final form factors for thesematerials the most accurate and detailed that are cur-rently available.

MCMIS uses four problem description input decksand various cross-section tables, and it outputs severalimage files and their associated stochastic uncertainties.The code contains four source models, three detectorgeometries, and three digital detector types. These mod-els and all of their parameters are listed in one of theinput decks. The geometry of the object being imagedand a list of materials are listed in other input decks.The last input deck contains information for the MonteCarlo run, i.e., the number of histories, variance-reduction methods to use, etc. In addition to the image,this code also calculates for each geometry region four

tallies: the energy deposited, the total flux, the expo-sure, and the dose.

The models include both a parallel synchrotronsource and a divergent point source for the Senoscan.The synchrotron system uses a Fuji imaging plate~BaFBr0.85I0.15! detector, and the Senoscan uses a CsIcharge-coupled-device detector. The photon interactiontypes are coherent scatter, incoherent scatter, and thephotoelectric effect. Cross sections include elements upto calcium and are tabulated for energies between 1 and300 keV, appropriate for medical imaging. Implicit cap-ture and the last-flight estimator variance-reduction tech-niques are available as options. The code does modelpolarization effects in the scattering interactions, whichis required to simulate the synchrotron-based mammog-raphy system. Extensive comparisons with other calcu-lations of scattering, dose, and scanning-slot responseswere made5 to verify the accuracy of MCMIS.

A series of steps are taken to simulate the actual im-age. First, the fine-mesh unscattered image and the coarse-mesh scattered image are added together. Then, noise, inan amount corresponding to actual noise seen in real im-ages, is added to the image, and the MTF is applied. TheMTF of the synchrotron system accounts for the smear-ing that occurs when the image plate is read by the laserscanner and converted into a digital image. The MTF ofthe Senoscan accounts for the blur caused by the focalspot size and blur introduced by the detector system. Forpolyenergetic sources, multiple monoenergetic runs of thecode are made and then added together using the sourcespectrum as the weighting function. A comparison of im-ages of the American College of Radiography~ACR!phantom created by MCMIS and taken on the real syn-chrotron imaging system is shown in Fig. 2. The imagetaken with the synchrotron shows vertical streaks causedby slight deformities in the crystal used to select the sin-gle energy of the beam. Measured contrast for items inreal digital images and the Monte Carlo simulationsmatched very well. A more complete description of themethods, testing, and verification of both images and dosecalculations made by MCMIS have been presented in anearlier paper.5

III. DIFFERENTIAL SAMPLING

Monte Carlo has the ability to calculate derivativesof responses with respect to certain input parameters atthe same time it calculates the response itself. This iscalled differential sampling, and some uses of such de-rivatives include

1. Perturbation studies: Instead of running the MonteCarlo code multiple times, a base responser can be cal-culated as a function of the set of input parametersp. Ifthis code also calculates the derivatives of the response

Fig. 1. Coherent scattering distributions in water for 18-keV photons using atomic form factors~dashed line! and usingNarten’s7 measured molecular form factor~full line!. The in-coherent scattering~dash-dot line! is also displayed.

MAMMOGRAPHY IMAGE SIMULATION 105


with respect to each parameter]r0]pi , and the secondderivatives]2r0]pi ]pj , and so on, then the value ofthe response for a small perturbation ofp can be foundfor any set of parameters by a Taylor series expansion.Thus,

r ~p 1 dp! 5 r ~p! 1 (iS ]r

]piDdpi

11

2 (i(

jS ]2r

]pi ]pjDdpi dpj 1 {{{ . ~1!

In practice, the series has to be truncated after a few terms~usually after the second-order terms!, and therefore, itis only applicable for values of~p 1 dp! close top. Cor-related sampling calculates all of the perturbed cases alongwith the reference case; differential sampling calculatesthe perturbed cases outside of the Monte Carlo code. If anew perturbation case is to be investigated, no more coderuns would be required once the reference case and itsderivatives had been found.

2. Sensitivity studies: The derivatives]r0]pi can alsobe used to find the sensitivity of the response to uncer-tainties in the input parameters. For example, which crosssections affect the response the most and need to be knownwith the highest accuracy? Differential sampling can re-veal this.

3. Total error: The uncertainty expressed in mostMonte Carlo studies is only the stochastic uncertaintyfrom the calculation. This does not represent the propa-gated uncertainties from any of the input parameters. Withthe sensitivities found by differential sampling, the totaluncertainty can be stated for a Monte Carlo response.

Differential sampling can be applied to many problems,even reactor problems, and can also be used in conjunc-tion with variance-reduction techniques.

This section will describe the system used to calcu-late the derivatives of Monte Carlo–calculated responsesand how they were implemented in the image simulationcode. The descriptions in this paper start with the MonteCarlo games as they are practiced, as opposed to startingwith the Boltzmann transport equation as in the pioneer-ing papers by Rief.9–12 The approach presented here iswritten with the Monte Carlo practitioner in mind.

III.A. Monte Carlo Game Description

The response calculated by a Monte Carlo game issimply the average of the individual responses resultingfrom the many histories. For a game where multiple con-tributions to the response tally are made during a history,the final averageSr has the form

Sr 51

N (i51

N

(j51

J

cij )k51

j

Pik , ~2!

where each segmentj of history i 5 1 to N makes somecontributioncij to the response tally, weighted by theproduct of segment probability densities)k51

j Pik up to

Fig. 2. Images of the ACR phantom:~a! image measuredwith the synchrotron system at 18 keV with a 5-mm slot sizeand~b! MCMIS Monte Carlo simulation.



that point in the history. The notation used here is similarto that used by Perel, Wagschal, and Yeivin.13 As pointedout by of Rief,9 these probability densities are either trans-port T or collisionC probabilities. Thus,

)k51

j

Pik 5 Ti1Ci1Ti 2Ci 2{{{ , ~3!

ending in either an absorption or escape in theJ’th seg-ment. Of course, the quantitiesPik are not explicitly cal-culated and multiplied together to form the product butare instead the result of the various stochastic choicesmade during the history ofJ segments.

An example of a game like this would be the calcu-lation of flux by the path-length estimator. Each time theparticle crosses a region, the path length~the contribu-tion cij ! multiplied by the product of the segment prob-abilities up to this point)k51

j Pik is added to a subtally.The product is equal tow, the current weighta of the par-ticle ~or 1 if no variance-reduction methods have beenemployed!. As the history continues, if the particle againcrosses the same region, the new path length multipliedby the current weight is added to the subtally. At the endof the history, the subtally value is added to the main re-sponse tally, and the square of the subtally is added tothe variance tally.

The foregoing game description can also be used forgames where splitting or point detectors are being used.When a particle is at a given point in the history, the con-tribution to a point detector or other tally is made, andthen the history continues on. The methods to calculatederivatives for these biased Monte Carlo games are thesame as for the basic games.

III.B. First Derivatives

The derivative of the final Monte Carlo response withrespect to a problem parametera is

]

]aSr 5

]

]aF 1

N (i51

N

(j51

J

cij )k51

j

PikG ~4!

51

N (i51

N

(j51

J Scij )k51

j

PikD3 H 1

cij

]

]acij 1 (

k51

j 1

Pik

]

]aPikJ . ~5!

This form contains the score made to the response sub-tally cij )k51

j Pik , multiplied by a term containing the rel-ative derivative of the contribution and the sum of therelative derivatives of the segment probability densities.With each path segment, the response for the segmentrij 5 cij )k51

j Pik ~computed asrij 5 wcij , wherew is theweight! is added to the response subtally. At the sametime, the subtallyqa

i for the derivative is also updatedwith

qai 5 (

j51

J

~rij ! H 1

cij

]

]acij 1 taijJ , ~6!

where taij is the current value of the relative segmentprobability derivative accumulator, up to this segment,

taij 5 (k51

j 1

Pik

]

]aPik . ~7!

The real key is then finding the expressions for relativederivatives~10Pik!~]0]a!Pik for both the collision andtransport kernels for every parametera and the relativederivative of the contributions~10cij !~]0]a!cij .

At the end of the history, when the response subtallyis added to the main response tally and its square is addedto its associated variance tally, each derivative subtallyis added to the main derivative tally, and its square isadded to a variance tally for the derivative. Let

Aa 5 ( qai ~8!

and

Ba 5 (~qai !2 . ~9!

At the end of the Monte Carlo calculation,Aa is usedfor finding the derivative~the sensitivity! with respect toparametera, andBa is used for finding the standard de-viation of the derivative:

]

]aSr 5

1

NAa ~10!

and

s] Sr0]a 5 ! 1

N F 1

NBa 2 S ]

]aSrD2G . ~11!

III.C. Double Derivatives

Derivatives with respect to any combination of twoparameters~including second derivatives! of the MonteCarlo responseSr can also be calculated. The double par-tial derivative of the response for the game described ear-lier with respect to the parametersa andb is

aThe photon weight is the probability of the photon beingat this point in the simulation had the simulation been an ana-log one. It is the ratio of the true probability distribution to thebiased probability distribution~each evaluated at the value ofthe sampled quantity!.



]

]b

]

]aSr 5

]

]b

]

]aF 1

N (i51

N

(j51

J

cij )k51

j

PikG ~12!

51

N (i51

N

(j51

J FScij )k51

j

PikDH(k51

j 1

Pik

]2

]a]bPik 2 (

k51

j S 1

Pik

]Pik

]a DS 1

Pik

]Pik

]b D1 S(

k51

j 1

Pik

]

]aPikDS(

k51

j 1

Pik

]

]bPikD1

1

cij

]2

]a]bcij

1 S 1

cij

]

]acijDS(

k51

j 1

Pik

]

]bPikD 1 S 1

cij

]

]bcijDS(

k51

j 1

Pik

]

]aPikDJG . ~13!

This can be more conveniently expressed as

]

]b

]

]aSr 5

1

N (i51

N

(j51

J F~rij ! Ht1,a,bij 2 t2,a,b

ij 1 taij tbij 11

cij

]2

]a]bcij 1 taij

1

cij

]

]bcij 1 tbij

1

cij

]

]acij JG , ~14!

where the two new accumulators~types 1 and 2! are

t1,a,bij 5 (

k51

j 1

Pik

]2

]a]bPik , ~15!

t2,a,bij 5 (

k51

j S 1

Pik

]Pik

]a DS 1

Pik

]Pik

]b D , ~16!

andtaij andtbij are already being kept for the calculation ofthe first derivative. During the history, every time the re-sponse subtally is updated, the subtally for each doublepartial derivative is updated:

qa,bi 5 (

j51

J F~rij ! Ht1,a,bij 2 t2,a,b

ij 1 taij tbij 11

cij

]2

]a]bcij

1 taij1

cij

]

]bcij 1 tbij

1

cij

]

]acij JG ~17!

using the current values of thet1,a,bij , t2,a,b

ij , taij , and tbij

accumulators.At the end of the history, when the response subtally

is added to the main response tally, each double partialderivative subtally is added to the main double partialderivative tally, and its square is added to the derivativevariance tally:

Aa,b 5 ( qa,bi ~18!

and

Ba,b 5 (~qa,bi !2 . ~19!

Then, at the end of the game,Aa,b is used for finding thedouble derivative, andBa,b is used for calculating the vari-ance of the double derivative.

The relative derivatives of the contributions and ofthe transport0collision kernels have not been discussedyet since they are problem dependent. The easiest way

to understand these is through an example. Most prob-lems will have similarT andC kernels and the contri-butions depend on what responses~tallies! one is tryingto determine.

More details on the foregoing differential samplingmethods and more examples of implementation into sim-ple problems are available elsewhere.14

IV. APPLICATION TO THE IMAGING CODE

MCMIS-DS, an extension of MCMIS for differentialsampling, provides the same four tallies~responses! foreach geometry region, namely, the energy depositedEd, thetotal fluxf, the exposureX, and the doseD. Of course, ex-posure is reported only when the region happens to con-sist of air. The parameters that one might be interested ininvestigating include the densityr of a region and its par-tial or total cross sectionsm 5 m phot1 mcoh1 minc.

Because cross sections are actually energy-dependent,trying to find the derivative of a response, such as energydeposited in a region, with respect to a particular crosssection could only be done at exactly one specific en-ergy. Another way to look at this problem is this: Sup-pose one did calculate the derivative of responser withrespect to cross sectionp. In calculating a truncated Tay-lor series to do a perturbation study,

r ~ p 1 dp! 5 r ~ p! 1 S ]r

]pDdp 1

1

2S ]2r

]p2D~dp!2

1 {{{ , ~20!

one would find that describingdp would be difficult sinceit is a function of energy and the responser is only ascalar.

The way MCMIS-DS overcomes this problem is bydefining the total macroscopic cross section of a regionas the weighted sum of the three interaction cross sections



for photoelectric effect, coherent scatter, and incoherentscatter. That is,

m~E! 5 aphotm phot~E! 1 acohmcoh~E! 1 aincm inc~E! ,

~21!

where the constantsai are all equal to 1 for an unper-turbed case. The code then finds the derivative of the re-sponses with respect to one of theai constants. This way,the energy dependence problem is avoided, and pertur-bations can be cast in a form of one partial cross sec-tion’s increasing~aphot going from 1 to 1.1! and anotherpartial cross section’s decreasing~acoh going from 1 to0.9!. This system is also somewhat easier for the usercompared to finding derivatives with respect to relativecross sections~mi0m! as others do,9 since there is no am-biguity about how the other cross sections are changingwhen a study of one of the interaction cross sections ischanging. With the foregoing system, it is obvious thatthe total cross section does change. So, MCMIS-DS hasfour parameters for derivatives: the density and the threereaction-type multipliers—aphot, acoh, andainc—for eachregion.

MCNP ~Ref. 15! has the ability to calculate pertur-bations on the weight fraction of a certain element in acompound. Since MCMIS-DS evaluates cross sectionsby region at the beginning, this type of perturbation couldnot be included. Also, the code uses measured molecularform factors for coherent scatter, so the effect of a weightfraction change for a given element in a molecular ma-terial could not be found since the form factor could bemeasured only at the nominal weight fraction values. Oneshould also be careful here since increasing one weightfraction implies that the others are also decreasing. It isnot clear from the MCNP manual how this is taken intoconsideration.

For a problem ofR regions, four tallies in each re-gion ~the energy deposited, the total flux, the exposure,and the dose!, and four parameters in each region tovary, there are 4R responses, 16R2 first derivatives, and8R2~4R1 1! double derivatives. For a small problem often regions, this works out to more than thirty thousandvalues, with thirty thousand associated uncertainties. Thisamount of information would be too much for any user,so MCMIS-DS does not keep track of every derivative.

MCMIS-DS allows the user to select one regionwhere the four tallies are the only four responses inves-tigated. Derivatives and double derivatives are found withrespect to the four variables in one or two regions~fouror eight total parameters!. This way, only the derivativesof interest are calculated and stored by the program, whichsaves time and disk space. These regions are listed in aseparate input file that is read by the program with theother input decks for geometry, cross section, etc.

The unscattered image is a matrix of tallies that couldnumber in the millions, depending on the number of pix-els. The coarse-mesh unscattered image can also be

viewed as a large matrix of tallies. Again, for time andstorage reasons, only a single first derivative and a sin-gle second derivative of the image are found with re-spect to only one of four parameters of one region. Thederivatives of the unscattered images are calculated withthe analytic calculation of the unscattered image. The de-rivatives of the scattered images are calculated by theMonte Carlo differential-sampling methods described inthe this paper.

IV.A. Derivatives of the Kernels

The Monte Carlo game consists of two kernels: trans-port T and collisionC. These are expressed as probabil-ity densities and are sampled by the Monte Carlo game.

The transport kernel is expressed as

T 5 rM mM expS2( rmmmsmD ~22!

for a particle that crosses many regionsm with a pathlength of sm in each and finally interacts in regionM.Using the notation from Sec. IV for the energy-dependentmass attenuation coefficients, the total mass attenuationcoefficient in each region is

mm 5 amphotmm

phot1 amcohmm

coh1 aminc mm

inc , ~23!

where the values of theami ’s are 1.

The relative derivative of the transport kernel is thenfound to be

1

T

]T

]rm

5 H2mmsm mÞ M

2mmsm 1 10rm m5 M ,~24!

with respect to any of the region densities. With respectto one of the cross-section coefficientsam

i , where the in-teraction typei is photoelectric, coherent, or incoherent,the relative derivative of the transport kernel is

1

T

]T

]ami 5 H2rmmm

i sm mÞ M

2rmmmi sm 1 mm

i 0mm m5 M .~25!

The collision kernels for interaction typei is ~in thematerial where the reaction is taking place!

Cphot 5aphotm phot

m~26!

Ccoh 5acohmcoh

mPcoh~V r V' ! ~27!

Cinc 5aincm inc

mPinc~V, E r V', E ' ! , ~28!

where thePi terms represent the scattering distributions forphotons of direction and energy ofV, E to scatter to di-rection and energy ofV', E ' . The relative derivative of thecollision kernel for reaction typekwith respect to the cross-section multiplierai is ~suppressing the subscriptm!

1

Ck

]Ck

]ai 5 H2m i0m i Þ k

2m i0m 1 10ai i 5 k .~29!



Since none of the collision kernels depend on the density of the material,~10Ck!~]Ck0]r! 5 0. Derivatives of thecollision kernels in one region with respect to a parameter in another region are also 0.

The double partial derivatives of the transport kernel can also be found to be

1

T

]

]rm

]

]rn

T 5 5mmsmmnsn mÞ M, n Þ M

mmsmmnsn 2mnsn

rm

m5 M, n Þ M

mmsmmmsm 2 2mmsm

rm

m5 n 5 M ,

~30!

1

T

]

]ami

]

]anj T 5 5

rmmmi smrn mn

j sn mÞ M, n Þ M

rmmmi smrn mn

j sn 2mm

i

mm

rn mnj sn m5 M, n Þ M

rmmmi smrmmm

j sm 2 2mm

i

mm

rmmmj sm m5 n 5 M ,

~31!

and

1

T

]

]ami

]

]rn

T 5 5rmsmmnsn mm

i mÞ M, n Þ M, mÞ n

rmsmmnsn mmi 2 smmm

i mÞ M, n Þ M, m5 n

rmsmmnsn mmi 2

mnsn

mm

mmi m5 M, n Þ M

rmsmmnsn mmi 2

rmsm

rn

mmi mÞ M, n 5 M

rmsmmmsmmmi 2 3smmm

i 1mm

i

rmmm

m5 n 5 M ,

~32!

and the double partial derivatives of the collision kernels are~again, suppressing the subscriptm!

1

Ck

]

]ai

]

]a j Ck 5 52

m i

m

m j

mi Þ k, j Þ k

2m i

m

m j

m2

m j

aimi 5 k, j Þ k

2m i

m

m j

m2 2

m i

aimi 5 j 5 k .

~33!

Any double derivative of any of the collision kernels withrespect to density is zero.

IV.B. Derivatives of the Tally Contributions

The relative derivatives of the tally contributions mustalso be found with respect to each of the parameters. Eachof the four responses and the point-detector contributionfor the scattered image calculation will be considered inturn.

IV.B.1. Energy Deposited

The first response is the energy deposited in a re-gion. In an analog game, the contribution is justc 5 E,the current energy of the particle for a photoelectric ef-fect orc5 Ee, the energy of the Compton electron for an

incoherent scatter. The relative derivatives of these withrespect to the density of a region or any of the cross-section coefficients are zero. If implicit capture is beingused, the contribution at each interaction is thenc 5Eaphotm phot0m. The relative derivatives are then similarto that of the photoelectric collision kernel:

1

c

]

]rc 5 0 ~34!

and

1

c

]

]ai c 5 H2m i0m i [ @coh, inc#

2m i0m 1 10ai i 5 phot .~35!



The relative double derivatives are then

1

c

]

]ai

]

]a j c 5 52

m i

m

m j

mi andj [ @coh, inc#

2m i

m

m j

m2

m j

aimi 5 phot, j [ @coh, inc#

2m i

m

m j

m2 2

m i

aimi 5 j 5 phot .

~36!

IV.B.2. Total Flux

For flux computed by the path-length estimator, thecontribution to the tally isc 5 s0V, wheres is the pathlength through the region. This is not dependent on anyof the parameters used in this study, so the derivativesand double derivatives are all zero.

IV.B.3. Exposure

For exposure in a region, the path-length estimatoris also used. The contribution to the tally is the productof the path lengths, the current energyE, and the massattenuation of air~at STP! at that energy~men~E!0r!air ,giving c 5 ~e0w!~sE0V !~men~E!0r!air , wheree is theelectron charge andw is the amount of energy neededto ionize air. This is not dependent on any of the param-eters used in this study, so like with the total flux con-tribution, the derivatives and double derivatives are allzero.

IV.B.4. Dose

Because of the low X-ray energies, secondary elec-trons are not tracked. For the calculation of dose in a re-gion~energy deposited divided by the mass of the region!in an analog game, the contribution to the response tallyis simply c 5 Ed 0rV, whereEd is the energy deposited~currentE for photoelectric orEe for incoherent scatter!andV is the volume of the region. The derivatives of thecontribution with respect to parameters of the same re-gion are

1

c

]

]rc 5 2 10r ~37!

and

1

c

]

]ai c 5 0 . ~38!

The relative second derivative with respect to the den-sity of the region is

1

c

]2

]r2 c 5 20r2 . ~39!

If implicit capture is being used, at each interaction,the contribution of

c 5E

rV

aphotm phot

m~40!

is scored to the dose tally. The derivatives of this contri-bution with respect to parameters of the same region are

1

c

]

]rc 5 2 10r ~41!

and

1

c

]

]ai c 5 H2m i0m i [ @coh, inc#

2m i0m 1 10ai i 5 phot .~42!

The relative double derivatives are

1

c

]2

]r2 c 5 20r2 , ~43!

1

c

]

]ai

]

]a j c 5 52

m i

m

m j

mi andj [ @coh, inc#

2m i

m

m j

m2

m j

aimi 5 phot, j [ @coh, inc#

2m i

m

m i

m2 2

m i

aimi 5 j 5 phot ,

~44!



and

1

c

]

]ai

]

]rc 5 52

m i

rmi [ @coh, inc#

2m i

rm1

1

airi 5 phot .

~45!

IV.B.5. Point Detector

For creating the scattered image, at each interaction point, the code computes the probability of both a coherentand incoherent scatter to each pixel in the image. This is very similar to the point-detector scheme used in manycodes. The contribution to a pixel of areaA and normal vector[n, and located in direction<V' from the interaction siteand interaction typej ~coherent or incoherent scatter! occurring in regionM is

cj 5aM

j mMj

mM

P j ~V, E r V', E ' !expS2( rmmm' smD AV'{ [n

S( smD2 , ~46!

wheresm is the distance traveled through the various regions between the interaction site and the image pixel. Afterthe scatter, the energy may change, and the new mass attenuation coefficientsmm

' ~and each of their component crosssections! are evaluated at the new energy.

The derivatives of the contribution are

1

ck

]

]rm

ck 5 2mm' sm ~47!

and

1

ck

]

]amj ck 5 5

2rm~mmj !'sm mÞ M

2rm~mmj !'sm 2 mm

j 0mm m5 M, j Þ k

2rm~mmj !'sm 2 mm

j 0mm 1 10amj m5 M, j 5 k .

~48!

The second derivatives are

1

ck

]2

]rm2 ck 5 ~mmsm!2 ~49!

and

1

ck

]2

]~amj !2 ck 5 5

~ rm~mmj !'sm!2 mÞ M

~ rm~mmj !'sm!2 1 2Smm

j

mmD2

1 2mm

j

mm

rm~mmj !'sm m5 M, j Þ k

~ rm~mmj !'sm!2 1 2Smm

j

mmD2

1 2mm

j

mm

rm~mmj !'sm

2 21

amj Srm~mm

j !'sm 1mm

j

mmD m5 M, j 5 k .

~50!

The point-detector algorithm was only used in calculating the scattered image. Because of space limitations, onlythe derivative and second derivative of the image with respect to one parameter is found by MCMIS-DS. With thislimitation, cross derivatives such as~]0]am

j !~]0]rm!ck are not needed.



IV.C. Implementation

Implementing thedifferential-samplingscheme for theresponses~four tallies and the images! of the MCMIS codeis fairly simple. For every responser, a subtallyq~r ! is kept.At the end of each history, the subtallies are added to themain tallies, and the squares of the subtallies are added tothe main variance tallies. For each combination of re-sponser and parametera, a derivative subtallyq~r,a!is kept. Similarly, for every combination of response,parametera and parameterb, a second derivative sub-tally q~r,a,b! is kept.At the end of each history, these sub-tallies are added to the main derivative tallies, and theirsquares are added to the derivative variance tallies.

Independent of the responses, other accumulators arekept. For every parametera, the accumulatorta is kept asan element of an arrayt~a! for the current sum of the rel-ative derivatives of the kernels. For every pair of param-eters, the two types of second derivative accumulatorst1,a,bandt2,a,b are kept in two other arrays,t1~a,b! andt2~a,b!.These accumulators follow the photon path, similar to theweight of the photon.~Here we have dropped thei super-script denoting thei ’th history. It is clear that the same ac-cumulatorscanbeclearedand reusedwitheachhistory.Thej superscript has also been dropped since these accumu-late over each segment for the history.!

During a history, the accumulatorsta, t1,a,b, andt2,a,bare updated at every transport step and collision for everyparametera andb. The subtally for a particular responsederivative is updated only when the subtally for that re-sponse is updated. In general, whenever a subtally for a re-sponse is about to be updated, the following also occur:

1. Calculate the segment contribution to the tally~tallynumberZ! ascij and set the tally score as the photon weightmultiplied by the segment contributionscore5 wcij .

2. Calculate all relative derivatives and double de-rivatives ofcij .

3. Update the response subtallyq~Z!5q~Z!1score.

4. Update every derivative subtally with respect toevery parametera

q~Z,a! 5 q~Z,a! 1 score3 S 1

cij

]

]acij 1 t~a!D .

5. Update every second derivative subtally

q~Z,a,b! 5 q~Z,a,b! 1 score

3 St1~a,b! 2 t2~a,b! 1 t~a!t~b!

11

cij

]2

]a]bcij 1 t~b!

1

cij

]

]acij

1 t~a!1

cij

]

]bcijD .

Note that the current values up to this segment ofw, t~a!,t~b!, t1~a,b!, andt2~a,b! are used here.

At the end of a history, the response subtallies areadded to the main tallies, and the squares of the subtal-lies are added to the variance tallies. The same procedureis used for the first derivative subtallies and the double-derivative subtallies.

After all N histories of the simulation are complete,the final calculations can be performed. First, the finalresponses are found by dividing the tally by the numberof histories, and the associated standard deviation is found.Then, the derivatives of the responses are found by di-viding the derivative tallies by the number of histories,and then their standard deviations are calculated.

IV.D. Results

A simple test can be used to demonstrate that thisdifferential-sampling scheme is calculating the right val-ues of the derivatives. Simulations of a 1.5-cm-thick pieceofLucite ina20-keVsynchrotron imagingbeamweremadeat a set of values of the Lucite density or the values of thecoefficients in the cross sectionm5aphotmphot1acohmcoh1aincminc. Within each set, the other parameters were keptconstant at their nominal values. Each simulation calcu-lated the four responses~Ed , f, X, andD! in the Luciteslab—the derivatives with respect to each parameter andthe second derivatives with respect to each parameter.

By comparing the derivatives calculated by differen-tial sampling to the derivatives estimated from the re-sponses calculated by the separate runs of a set, the validityof the differential sampling can be shown. This is done inFig. 3 for the dose with respect to the Lucite density and inFig. 4 for the energy deposited with respect to the coeffi-cient for photoelectric absorption. Figures 3a and 4a showthe response at the various values of the parameter. Fig-ures 3b and 4b show the derivative of the response with re-spect to the parameter calculated by differential sampling~the points! and the estimate of the derivative~line! by tak-ing the derivative of a polynomial fitted to the Monte Carloresponses. Figures 3c and 4c show the second derivativeof the response with respect to the parameter calculated bydifferential sampling~the points! and the estimate of thederivative~line! by taking the derivative of a polynomialfitted to the Monte Carlo first derivatives.

The variance of the derivatives calculated by thedifferential-sampling method outlined in this paper werealso checked to see if they correctly predicted the sto-chastic variance. The variance found in one hundred sep-arate Monte Carlo runs using different random seeds didindeed match the average of the variances predicted bydifferential sampling for those runs.

V. MCMIS-DS WITH A BREASTLIKE PHANTOM

To show how differential sampling can be appliedto mammography problems, a series of simulations and



calculations were carried out for a 4-cm-thick com-pressed breast in the Fischer Senoscan imaging system.The two questions that will be answered by the code areas follows. First, what is the sensitivity of the dose to thebreast tissue with respect to the parameters of the breasttissue and the tumor tissue? Second, what density in-crease must a tumor have over that of ordinary breasttissue to be visible in the image? Both of these questionsare important in medical physics, and they can be an-swered by using MCMIS-DS.

The reference model for the simulations included acompressed breast, consisting of ICRU 50050 water0lipid whole breast material,16 with a spherical tumor ofthe same material in the center. The breast is covered witha 0.4-cm-thick skin layer that is also made of this mate-rial. A cross section of this geometry is shown in Fig. 5,which also shows very thick skin on the sides of the breast.This was done so that doses reported for the simulationsinclude only the breast material in the center portion of

Fig. 3. ~a! Dose~rad0photon! in a slab of Lucite for dif-ferent values of the density,~b! ]D0]r, and~c! ]2D0]r2. Thepoints with error bars are from independent Monte Carlo cal-culations. The lines in~b! and ~c! are the derivatives of thecurve fit of the Monte Carlo data points from~a! and ~b!,respectively.

Fig. 4. ~a! Energy deposited~keV0photon! in a slab of Lu-cite for different values of the photoelectric cross-section co-efficient, ~b! ]Ed 0]aphot, and~c! ]2Ed 0]~aphot!2. The pointswith error bars are from independent Monte Carlo calcula-tions. The lines in~b! and~c! are the derivatives of the curvefit of the Monte Carlo data points from graphs~a! and ~b!,respectively.



the breast. Skin dose tolerance limits are much higherthan for the breast tissue.

In addition to the geometry shown in Fig. 5, the Fis-cher scanning-slot detector system was used in the sim-ulation. The measured coherent scattering form factor ofhuman breast tissue8 was used for the breast, skin, andtumor regions. For the images, a pixel size of 100mmwas used for the fine-mesh images~unscattered, first de-rivative, and second derivative!, and a pixel size of 0.5 cmwas used for the coarse-mesh images~scatter and itsderivatives!.

To simulate a 25-kVp mammogram on the Seno-scan, 20 monoenergetic simulations were made from 6to 25 keV, at every 1 keV. The X-ray spectrum for a tung-sten anode tube was taken from Boone, Fewell, and Jen-nings17 and used as a weighting function to add togetherthe monoenergetic images, the dose information, and thesensitivity information.

For each monoenergetic code run, an average of6.5 min on a Sun Ultra 60 was required to create the threefine-mesh unscattered images. For the coarse-mesh scat-tered images, an average of 2.25 h was required. Thescattered-to-primary~S0P! ratio for this simulation of a4-cm compressed breast was 20% for most of the breast,with lower values toward the edges of the irradiation field.With this large anS0P ratio, the times for the scatteredimage calculations were selected to reduce the uncer-tainty in the polyenergetic scattered image to,2% overmost of the image. In total, for one polyenergetic Seno-scan simulation, 48 h of computer time on a Sun Ultra 60were required.

V.A. Sensitivity of Breast Region Dose

To answer the first question about the sensitivity ofthe dose in the breast tissue, each monoenergetic run ofMCMIS-DS found the derivatives of all of the tallies forthe breast region with respect to all of the parameters ofthe breast region and the tumor region. The results of theseruns were then added together with the same source spec-trum weighting function to find the sensitivities for thepolyenergetic spectrum. In addition to the tallies for thebreast region, 32 first derivatives and 256 double deriv-atives were found. A few of the quantities calculated by

MCMIS-DS for the combined 25-kVp spectrum are listedin Table 1.

From these values, the sensitivities of the breast doseto the different parameters were found. These sensitivi-ties are displayed in Table II, and from them, it is clearthat the dose in the breast region is much more sensitiveto the parameters of itself than to the parameters of thetumor region. Further analysis shows that the dose is moresensitive to the breast tissue density~or its total crosssection! and the photoelectric cross section than to eitherof the scattering cross sections.

The foregoing information can also be used in prop-agating errors. The 0.15% relative uncertainty reportedin Table I for the dose to the central portion of the breastis the stochastic error only. Using the value for~ r0D!~]D0]rbreast!, one could determine that a 1% uncertainty inthe breast tissue density gives an;0.64% uncertainty inthe dose value. A similar 1% uncertainty in the tumordensity gives only a 0.00067% uncertainty in the dose,which makes sense considering how little the density ofa small tumor should affect the absorbed dose in the breast.These uncertainties could then be combined with the sto-chastic uncertainty to give the total uncertainty in the dosevalue.

V.B. Tumor Density and Visibility

For the second question of what density difference acertain size tumor must have in order to be visible in theimage, the code was instructed to find the derivatives ofthe images with respect to the density of the tumor re-gion. These were calculated at the same time as the fore-going sensitivity figures.

The runs computed the images from a breast that con-tained a tumor of the same density, which is referred to

Fig. 5. Geometry of the compressed breast simulationusing the Fischer Senoscan. The breast support and compres-sion paddle hold the breast tissue to a 4-cm thickness. Only thecenter portion is used for the dose calculations; the rest is con-sidered skin.

TABLE I

A Few of the Tallies and Derivatives for the Breast Regionat 25-kVp Calculated by MCMIS-DS

Quantity~per source photon! Value

Energy deposited,E ~keV! 3.6436 0.15%Total flux, f ~cm22! 0.0013036 0.16%Dose,D ~rad! 1.975E213a 6 0.15%

]D0]r ~cm30g! 21.372E2136 0.24%]D0]aphot ~rad! 6.401E2146 0.3%]D0]acoh ~rad! 2.288E2176 270%]D0]ainc ~rad! 1.776E2156 4.2%

]2D0]r2 ~cm60g2! 28.584E2146 0.18%]2D0]aphot

2 ~rad! 27.188E2146 0.18%]2D0]acoh

2 ~rad! 3.93E2156 0.54%]2D0]ainc

2 ~rad! 6.76E215 6 0.6%

aRead as 1.9753 10213.



here as the reference case. Obviously, a tumor of the samecomposition and density as the surrounding tissue wouldnot be visible in the images. But, MCMIS-DS also com-puted the first and second derivatives of the image withrespect to tumor density. For a 1-cm tumor, both sets ofMonte Carlo unscattered and scattered images are shownin Fig. 6. From these images, an image can be createdfrom a Taylor series expansion for a tumor of any den-sity close to the reference case. Of course, real clinicalimages would not have such a smooth background asthe tissue structure of glands, and fat would be visibleand would somewhat hide the tumor. Ignoring this factfor the moment, this example is still useful in showingan application of differential sampling and in providinguseful information for system designers and medicalphysicists.

For a 1-cm-diam tumor, the images shown in Fig. 6were combined in a truncated Taylor series for six dif-ferent tumor densities:

I ~ r 1 dr! 5 I ~ r0! 1 S ]I

]rDdr 1

1

2S ]2I

]r2D~dr!2 ,

~51!

r0 5 0.92 g0cm3 , ~52!

and

dr [ r0 3 @0.01, 0.015, 0.02, 0.025, 0.03, 0.05# .

~53!

These images were then processed by adding 0.5%relative noise~typical in Senoscan images! and applyingthe MTF. The final six images are shown in Fig. 7. Fromthis series of images, one can judge what level of densitydifference is required before the tumor is visible. Clearly,

at a 2.0% density increase over that of the surroundingtissue, the tumor is visible, and different observers mayargue about visibility at the lower-density increases.

Series of images for smaller tumor sizes, 0.5 cm and0.25 cm diameter, are shown in Figs. 8 and 9. The 0.5-cmtumor is not visible until at least a 2.5% increase in tu-mor density over tissue density~again, other observersmay argue!. The smallest tumor~0.25 cm! is not visibleuntil a density increase of 5% over that of normal tissue.This kind of information is very useful in comparing dif-ferent mammography systems or in optimizing the spec-trum or other system parameters for a single system.

The real benefit of the differential sampling and Tay-lor series approach is the ability to return to the problemlater and add another value ofdr without running anymore Monte Carlo calculations. Once the reference caseimage and its derivatives have been calculated, any num-ber of images can be made with different tumor densi-ties. This is one big advantage of differential samplingover correlated sampling. In fact, since the polyenergeticspectrum was made of the weighted sum of monoener-getic spectra, the spectrum could be changed, and thestudy could be repeated without running any more MonteCarlos.

Similar perturbation calculations can be made forthe tallies in the breast region with the output fromMCMIS-DS that was shown in Tables I and II. For anychange in the parameters of either the breast tissue orthe tumor~or both simultaneously!, one could calculatethe new quantities.

VI. SUMMARY

We have formulated and implemented differentialsampling in a Monte Carlo code specifically written to

TABLE II

Sensitivities for the Breast Region Calculated from the Doses and Derivatives Found Using MCMIS-DS

Parameters in the

Quantity m5 breast m5 tumor

rm

D

]D

]rm

20.63906 0.28% 26.708E24a 6 5.8%

amphot

D

]D

]amphot 0.32416 0.33% 28.066E24 6 3.4%

amcoh

D

]D

]amcoh 1.158E24 6 270% 2.508E2056 50%

aminc

D

]D

]aminc 8.993E23 6 4.2% 8.15E2056 23%

aRead as26.7083 1024.



Fig. 6. Monte Carlo images created by MCMIS-DS for the breast0tumor problem:~a! The unscattered image,~b! the scat-tered image~c! and~d! the first derivative with respect to tumor density~unscattered and scattered!. The second derivatives arealso shown in panels~e! and~f !.



Fig. 7. Six images created by a truncated Taylor series for a 1-cm tumor. Each panel is labeled with the percentage increasein tumor density over that of the breast tissue. Noise and the system MTF have been added to these images, simulating what theFischer Senoscan would see for a 4-cm compressed breast.



Fig. 8. Six images created by a truncated Taylor series for a 0.5-cm tumor. Each panel is labeled with the percentage increasein tumor density over that of the breast tissue. Noise and the system MTF have been added to these images.



Fig. 9. Six images created by a truncated Taylor series for a 0.25-cm tumor. Each panel is labeled with the percentageincrease in tumor density over that of the breast tissue. Noise and the system MTF have been added to these images.



simulate two new and advanced digital mammographysystems. The code uses experimentally measured coher-ent scattering angular distributions of X rays in mammo-graphically relevant materials8 and simulates images on20483 2048 pixels and also calculates the exposure andthe absorbed dose to the sample.5 A variety of variance-reduction schemes were used to make the image simula-tion possible on standard computing platforms. Thesimulated results showed good fidelity in their compar-ison to experimental benchmarks on both systems.

Since it is a perturbation technique, differential sam-pling can reveal a great deal of information without add-ing much computational cost. Its application to MonteCarlo problems with variance-reduction methods is nomore difficult than to analog Monte Carlo simulations.Also, responses to perturbations of several system pa-rameters can be calculated simultaneously. Since the de-rivatives of responses and their variances are tallied, theresponses for other variations in the parameter values canbe computed at a later time without having to re-run thecode. This is a considerable advantage over the methodof correlated sampling. The methods presented in this pa-per make the implementation of differential samplingmuch easier for a Monte Carlo code developer when com-pared to the descriptions presented in the past in literature.

Our code, MCMIS-DS, gives the user considerablygreater flexibility in exploring mammography images thatcan be expected for parameter changes than any standardcode could give. Two examples are shown as demonstra-tions of the implementation of differential sampling. Inthe first example, we have shown how to formulate dif-ferential sampling for a dose-related problem involvingfour dose-related tallies for perturbations in sample den-sity and cross sections. The main thrust here has been todemonstrate that the response derivatives calculated bydifferential sampling and their variance agree exactly withthe numerical derivatives of the response and their sto-chastic variance. In the second example problem, we in-vestigated two medically relevant questions, namely, thesensitivity of the X-ray dose to the breast as a function oftumor and normal tissue parameters and the discernibil-ity of tumors of different sizes in an image of the breastas a function of tumor density in the case of one of thetwo mammography modalities~Fischer Senoscan!.

It should be stressed that the MCMIS-DS code itselfis not as important a part of this paper as the ideas usedin its development. The description of differential sam-pling and its application to the mammography image sim-ulation package are expected to provide others with theability to adapt and implement differential sampling fortheir own applications.

Concerning mammography studies, it has been shownthat a tool with the capabilities of MCMIS-DS can pro-vide useful information. In sensitivity analysis and per-turbation studies, these tools are very useful in comparingimaging systems and in designing and optimizing singlesystems. In the future, the concepts of the MCMIS and

MCMIS-DS package will be refined and built into a toolfor the mammographic imaging community.

Basic research into differential sampling also needsto continue. Two parameters were not explored in thisproject, and they are very important to medical imaging:the source energy of the photonsE0 and the physical di-mensionssof regions. Similar types of parameters as thesetwo are also important in many other areas, and it wouldbe of great benefit if differential sampling could be usedfor perturbations of these parameters. More research isrequired to do one of the following: either show how dif-ferential sampling can calculate]0]s and]0]E0 or showconclusively that these derivatives cannot be calculated.

ACKNOWLEDGMENTS

Research was carried out in part at the National Synchro-tron Light Source, Brookhaven National Laboratory, which issupported by the U.S. Department of Energy~DOE!, Divisionof Materials Sciences and Division of Chemical Sciences. Manythanks to W. C. Thomlinson and Z. Zhong for allowing thiswork to be done on beamline X15A.

R. E. Johnston of the University of North Carolina, ChapelHill, provided assistance in both the synchrotron imaging andSenoscan imaging aspects of this study. We thank him for allof his valuable help.

The first author~D.E.P.!, at the time a doctoral student,was supported by R. E. Johnston through his U.S. Army Med-ical Research and Material Command Breast Cancer ResearchGrant. He was also supported by a Nuclear Engineering Edu-cation Research Grant from the DOE Idaho Operations Office.

Thanks also to the referees who helped make this a betterpaper.

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